How can I refinish these mid-century dining chairs?

Yikes, Kathryn! That's a lot of money! If the wood is in good shape and the stain/ varnish isn't chipped or peeling anywhere, I would leave it. There are good wood cleaners you can use to remove that sticky feel and you can easily recover the chair seats yourself. They usually unscrew from the bottom. Then you would just remove the old fabric and add new. If the chair pads are flat and need replacing you can purchased a good foam for it. And to make the job easier, I would get a power staple gun, it's well worth the investment for the job, if you don't have one. All in all, it won't cost you anywhere near the price you've been quoted and it really is a simple DIY. Hope that helps!

https://www.youtube.com/watch?v=wCcr8MYYBHo

It looks as if you can remove the seats and recover yourself. Look underneath the seats to see if there are screws you can remove and then the seats should just pop out. The material is more than likely just stapled on as it looks like that material is not original to the age of the chair. Sometimes people recover the seats right over old material so you may be able to use one piece of the material as a pattern for new. Recovering these seats should not be difficult and should not cost a huge amount. I think when you mean the chairs are "sticky" you are referring to the wood. Have you tried cleaning the chairs with Murphy's Oil Soap--it may remove the stickiness. I would bet it is the arms that are sticky. You could go over the arms with some fine steel wool to remove any stickiness if it doesn't come off after you clean it well. You may be able to get by with just touching up the arms with a matching stain and then finish with a poly. The rest of the chair looks to be in good shape--it's hard to tell in a picture. Beautiful chairs! I would try to leave them in their original state as much as possible as long as they are solid not wobbly.

It is SO easy to reupholster a chair!!! You'll just need some chair padding, fabric and a staple gun... here is a chair I redid for my daughters room, but you'd follow the same process for yours (just pick out your paint color, fabric, etc.): https://www.curlycraftymom.com/2015/05/making-good-blog-train-desk-chair-revamp.html

https://www.hometalk.com/search/posts?filter=upholster%20chairs

sticky problem is more than likely caused by the finish itself failing. Drying oils in the varnishes eventually destroy themselves and the sticky surface you describe is the result. Lacquers can suffer a problem caused by plasticizers migrating through the resins. One way or another, your finish surface is done. best advice would be to clean it chemically to a surface that is not sticky,sand it down completely back to bare wood, first with 80 grit and then 120 and 220 grits, it'll provide the best finish overall. Re-stain or recolor to your satisfaction, then seal with shellac to prevent tannin bleed thru

That price is OUTRAGEOUS! I recommend you get some CitriStrip which is non-toxic and works great. Slather it all over the chairs, cover with plastic for it to work even better and then scrape the old finish off with a flexible putty knife and drop the gunk directly into a garbage can. It's not difficult!

Once stripped, go over the chair with sandpaper. Your grit depends on the condition once stripped but you can probably get away with starting at 120 grit from the looks of the chairs. They are great!

Clean up the whole thing by wiping down with mineral spirits and clean towel, let dry thoroughly. Restain as desired and seal with polyacrylic as it does not yellow.

Reupholstering is simple and I know you can do it!

The project is cheap, easy on the scale of DIY's and will give you a great sense of satisfaction once done. Go for it!!!